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Introduction to Normed *-Algebras and their Representations, 7th ed

Marco Thill

TL;DR

Thill's monograph develops the spectral theory of normed $*$-algebras and their representations, culminating in the enveloping $C^*$-algebra construction that unifies algebraic and analytic approaches. It first covers the classical spectral theory for Banach $*$-algebras (including the Shirali–Ford theorem) and then develops a representation-theoretic framework via the GNS construction for general normed $*$-algebras, allowing a transfer of results through the enveloping $C^*$-algebra. The text then establishes the normed spectral calculus and a robust Spectral Theorem for commutative $*$-algebras, with disintegration results akin to Bochner-type theorems obtained by measure-theoretic means. It also provides foundational tools such as Stone–Weierstrass, automatic continuity, boundary results for spectra, and a detailed treatment of Hermitian Banach $*$-algebras, culminating in the theorem that $C^*$-algebras are precisely the natural, Hermitian normed $*$-algebras suited for representation theory.

Abstract

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C^*-algebra of a general normed *-algebra.

Introduction to Normed *-Algebras and their Representations, 7th ed

TL;DR

Thill's monograph develops the spectral theory of normed -algebras and their representations, culminating in the enveloping -algebra construction that unifies algebraic and analytic approaches. It first covers the classical spectral theory for Banach -algebras (including the Shirali–Ford theorem) and then develops a representation-theoretic framework via the GNS construction for general normed -algebras, allowing a transfer of results through the enveloping -algebra. The text then establishes the normed spectral calculus and a robust Spectral Theorem for commutative -algebras, with disintegration results akin to Bochner-type theorems obtained by measure-theoretic means. It also provides foundational tools such as Stone–Weierstrass, automatic continuity, boundary results for spectra, and a detailed treatment of Hermitian Banach -algebras, culminating in the theorem that -algebras are precisely the natural, Hermitian normed -algebras suited for representation theory.

Abstract

This book treats: - spectral theory of Banach *-algebras, - basic representation theory of normed *-algebras, - spectral theory of representations of commutative *-algebras. A novel feature of the book is the construction of the enveloping C^*-algebra of a general normed *-algebra.

Paper Structure

This paper contains 63 sections, 498 theorems, 952 equations.

Table of Contents

  1. Spectral Theory of Banach 80$_{\mathrm{o}}$5280$_{\mathrm{o}}$55Algebras
  2. Normed 80$_{\mathrm{o}}$5280$_{\mathrm{o}}$55Algebras: Basic Definitions and Facts
  3. 80$_{\mathrm{o}}$5280$_{\mathrm{o}}$55Algebras and their Unitisation
  4. Normed 80$_{\mathrm{o}}$5280$_{\mathrm{o}}$55Algebras and their Unitisation
  5. The Completion of a Normed Algebra
  6. Some Pre80$_{\mathrm{o}}$55C80$_{\mathrm{o}}$5280$_{\mathrm{o}}$55Algebras and Lattices of Functions
  7. The Stone-Weierstrass Theorem
  8. Basic Properties of the Spectrum
  9. The Spectrum
  10. The Spectral Radius Formula: r8013783273
  11. Properties of r8013783273
  12. The Pt80341k Function r8013783303
  13. Automatic Continuity
  14. The Boundary of the Spectrum
  15. Square Roots of Invertible Elements
  16. ...and 48 more sections

Key Result

Proposition § 1.15

normal!element|)A $\ast$-algebra is commutative if and only if each of its elements is normal.

Theorems & Definitions (1139)

  • Definition § 1.1: algebra
  • Example § 1.2: $\mathrm{End}(V)$
  • Definition § 1.3: involution, adjoint
  • Definition § 1.4: algebra involution
  • Definition § 1.5: $\ast$-algebra
  • Example § 1.6: $\mathds{C}^{\,\text{\Small{$\Omega$}}}$
  • Example § 1.7: $\mathrm{B}(H)$
  • Definition § 1.8: the free vector space $\mathds{C}[S]$
  • Example § 1.9: the group ring $\mathds{C}[G]$
  • Definition § 1.10: $\ast$-semigroup
  • ...and 1129 more